help in solving by completing the square

Question

I am trying to solve 2x^2 + 3x =20 by completing the square.

 

I rewrote it to be 2x^2 - 3x + 20 = 0

 

then started with 2x^2/2 + 3x/2 + 20/10 =   x^2 - 3/2x + 10 = 0

 

how do I continue?

Answer 1

"I am trying to solve 2x^2 + 3x =20 by completing the square."

 

2x^2 + 3x = 20

 

Divide both sides by 2:

x^2 + (3/2)x = 10

 

We want to force a pattern that gives us a perfect square; i.e.,

(x-h)^2 = x^2 - 2hx + h^2.

 

So force our equation into having that pattern:

x^2 - 2(-3/4)x + (-3/4)^2 = 10 + (-3/4)^2

 

Replace the perfect square trinomial on the left

with a perfect square:

(x + 3/4)^2 = 10 + 9/16 = 169/16 = (13/4)^2

 

Take the square root of both sides (√(z^2) = |z|):

|x + 3/4| = 13/4

 

Solve for x:

x + 3/4 = ±13/4

x = -3/4 ± 13/4

x = (-3 ± 13)/4

x = -4 or 5/2

Answer 2

 

  2X^2 -3X + 20 =0

 

  2 ( X^2 -3/ 2 X + 10 ) = 0

 

   ( X ^2 - 2( 3/4 ) X  ) = -10

    

     ( X^2 - 2(3/4) X + 9/16) = -10 + 9/16

 

     ( X - 3/ 4 ) ^2 = -151/ 16 

 

     X - 3/4  = ± √151/4 i

 

       X = 3/4 ± i√151/4 

       

 

 

Answer 3

There are several ways to solve a quadratic equation by completing the square.

 

The approach I like best is to work with the equation in the form you had originally:

 

2 x² +3 x = 20             {the constant term on the right hand side}

 

From this we identify a as  a = 2  and b as   b= 3.      The number to add to both sides is  b²/4a =  9/8

 

This results in

 

2  x² + 3 x + 9/8  =   20 + 9/8 = 169/8

 

Next we divide the equation by 2 to get the coefficient of x² to be 1

 

  x²  + (3/2) x + 9/16 = 169/16

 

Because we added the b²/4a  , the left hand side will be perfect square.   It is easy to see that it is the square of (x + 3/4)    Thus we have

 

  (x + 3/4)^2 =  169/16

 

We next take the square root of both sides to get

 

  (x + 3/4) =   ± 13/ 4

 

So finally   x = -3/4  ± 13/4        or

 

x =   5/2     and x  = -4

 

With this general approach, the only thing to memorize is that the quantity to add is b²/4a